(x^3-x)dy/dx-(3x^2-1)y=x^5-2x^3+x

Jasonbradley

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Jun 17, 2024

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Solving the Differential Equation: (x^3 - x)dy/dx - (3x^2 - 1)y = x^5 - 2x^3 + x

This article will guide you through the process of solving the given first-order linear differential equation.

Understanding the Equation

The equation is a first-order linear differential equation because it involves the first derivative of y (dy/dx) and both y and its derivative appear linearly. The equation is of the form:

P(x) dy/dx + Q(x)y = R(x)

where:

• P(x) = x^3 - x
• Q(x) = -(3x^2 - 1)
• R(x) = x^5 - 2x^3 + x

Solving the Equation

We will use the method of integrating factors to solve this equation.

1. Finding the Integrating Factor:

The integrating factor is given by:

μ(x) = exp(∫(Q(x)/P(x)) dx)

In this case:

μ(x) = exp(∫(-(3x^2 - 1)/(x^3 - x)) dx)

We can simplify the integrand by factoring the denominator:

μ(x) = exp(∫(-(3x^2 - 1)/(x(x^2 - 1))) dx) = exp(∫(-(3x^2 - 1)/(x(x - 1)(x + 1))) dx)

Now we can use partial fractions to break down the integrand:

(-(3x^2 - 1)/(x(x - 1)(x + 1))) = (A/x) + (B/(x - 1)) + (C/(x + 1))

Solving for A, B, and C, we get:

A = 1, B = -2, C = 1

Therefore:

μ(x) = exp(∫(1/x - 2/(x - 1) + 1/(x + 1)) dx)

Integrating:

μ(x) = exp(ln|x| - 2ln|x - 1| + ln|x + 1|) = exp(ln(|x(x + 1)/(x - 1)^2|)) = |x(x + 1)/(x - 1)^2|

We can choose the positive sign for simplicity, since the integrating factor is used for multiplication.

2. Multiplying the Equation by the Integrating Factor:

Multiplying the original equation by the integrating factor:

[x(x + 1)/(x - 1)^2] [(x^3 - x)dy/dx - (3x^2 - 1)y] = [x(x + 1)/(x - 1)^2] [x^5 - 2x^3 + x]

3. Simplifying the Equation:

Notice that the left-hand side now becomes the derivative of a product:

d/dx [y(x(x + 1)/(x - 1)^2)] = [x(x + 1)/(x - 1)^2] [x^5 - 2x^3 + x]

4. Integrating Both Sides:

Integrating both sides with respect to x:

∫ d/dx [y(x(x + 1)/(x - 1)^2)] dx = ∫ [x(x + 1)/(x - 1)^2] [x^5 - 2x^3 + x] dx

This simplifies to:

y(x(x + 1)/(x - 1)^2) = ∫ [x(x + 1)/(x - 1)^2] [x^5 - 2x^3 + x] dx + C

5. Solving for y:

Finally, solve for y:

y = [(x - 1)^2/(x(x + 1))] ∫ [x(x + 1)/(x - 1)^2] [x^5 - 2x^3 + x] dx + C[(x - 1)^2/(x(x + 1))]

Conclusion

The solution to the differential equation is given by the above expression, where C is an arbitrary constant of integration. The integral on the right-hand side needs to be evaluated to get the explicit solution. This process demonstrates the power of the integrating factor method for solving linear first-order differential equations.